%%%%%%%%%%%% real, integer notation
\newcommand{\real}{{\mathbb R}}
\newcommand{\integer}{{\mathbb Z}}


% boldface characters
% macro by mung

\newcommand{\td}{\ensuremath{t_{\text{dl}}}}
\newcommand{\bx}{\bm{x}}
\newcommand{\bc}{\bm{c}}
\newcommand{\bbx}{\ensuremath{{\bm x}}}
\newcommand{\bbc}{\ensuremath{{\bm c}}}
\newcommand{\bu}{\ensuremath{{\bm u}}}

\newcommand{\bz}{\ensuremath{{\bm z}}}
\newcommand{\ba}{\ensuremath{{\bm a}}}
\newcommand{\be}{\ensuremath{{\bm e}}}
\newcommand{\br}{\ensuremath{{\bm r}}}
\newcommand{\by}{\ensuremath{{\bm y}}}
\newcommand{\bby}{\ensuremath{{\bm y}}}
\newcommand{\bbq}{\ensuremath{{\bm q}}}


\newcommand{\bp}{\ensuremath{{\bm p}}}
\newcommand{\bh}{\ensuremath{{\bm h}}}
\newcommand{\bP}{\bm{P}}
\newcommand{\bA}{\ensuremath{{\bm A}}}

\newcommand{\bF}{\bm{F}}
\newcommand{\bN}{\ensuremath{{\bm N}}}
\newcommand{\bbf}{\ensuremath{{\bm f}}}
\newcommand{\bG}{\ensuremath{{\bm G}}}
\newcommand{\bQ}{\ensuremath{{\bm Q}}}
\newcommand{\bH}{\ensuremath{{\bm H}}}
\newcommand{\bW}{\ensuremath{{\bm W}}}
\newcommand{\bJ}{\ensuremath{{\bm J}}}
\newcommand{\bg}{\ensuremath{{\bm g}}}
\newcommand{\bs}{\ensuremath{{\bm s}}}

\newcommand{\bbv}{\ensuremath{{\bm v}}}
\newcommand{\bbz}{\ensuremath{{\bm z}}}
\newcommand{\bbu}{\ensuremath{{\bm u}}}
\newcommand{\bbm}{\ensuremath{{\bm m}}}
\newcommand{\bbU}{\ensuremath{{\bm U}}}
\newcommand{\bw}{\bm{w}}
\newcommand{\bbr}{\ensuremath{{\bm r}}}
\newcommand{\bbalpha}{\ensuremath{{\bm \alpha}}}
\newcommand{\brho}{\ensuremath{{\bm \rho}}}


% \newcommand{\bblambda}{\mbox{\boldmath \ensuremath{\lambda}}}
% \newcommand{\blambda}{\mbox{\boldmath \ensuremath{\lambda}}}

% \newcommand{\bmu}{\mbox{\boldmath \ensuremath{\mu}}}
% %\newcommand{\bmu}{\mu}

% \newcommand{\bbeta}{\mbox{\boldmath \ensuremath{\beta}}}
% \newcommand{\btheta}{\mbox{\boldmath \ensuremath{\theta}}}
% \newcommand{\bv}{\ensuremath{{\mathbf v}}}
% %\newcommand{\reals}{\ensuremath{\mathbf{R}}}
\newcommand{\bR}{\bm{R}}
% \newcommand{\bw}{\ensuremath{{\mathbf w}}}
% % \newcommand{\bF}{F}


%%%%%%%%%%%% Sets, calligraphic characters...
\newcommand{\cR}{\ensuremath{\mathcal R}}
\newcommand{\set}[1]{\ensuremath{\mathcal #1}}
\renewcommand{\vec}[1]{\bm{#1}}
\newcommand{\mat}[1]{\bm{#1}}

\newcommand{\node}[1]{\ensuremath{{\tt #1}}}
\newcommand{\nn}[1]{\node{#1}}

\newcommand{\singleq}[1]{\text{`}#1\text{'}}

\newcommand{\mytitle}[1]{\medskip \noindent{\em #1} \smallskip}
\newcommand{\mytitlehead}[1]{\noindent{\em #1} \smallskip}

\newcommand{\separator}{
  \begin{center}
    \rule{\columnwidth}{0.3mm}
  \end{center}
}
\newenvironment{separation}{ \vspace{-0.3cm}  \separator  \vspace{-0.2cm}}
{  \vspace{-0.4cm}  \separator  \vspace{-0.1cm}}

%%%%%%%%%%%%%%%% other macros %%%%%%%%%%%%%%%%%%
\def\un{\underline}
\def\ov{\overline}
\def\Bl{\Bigl}
\def\Br{\Bigr}
% \def\bl{\bigl}
% \def\br{\bigr}
\def\lf{\left}
\def\ri{\right}
\def\st{\star}

\def\eg{{\it e.g.}}
\def\ie{{\it i.e.}}

%\newcommand{\eqdef}{{\triangleq}}

% \newcommand{\bi}{\begin{itemize}}
% \newcommand{\ei}{\end{itemize}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
% \newtheorem{algorithm}{Algorithm}[section]
\newtheorem{condition}{Condition}[section]
\newtheorem{assumption}{Assumption}[section]
\newtheorem{example}{Example}[section]
\newtheorem{remark}{Remark}[section]
\newtheorem{problem}{Problem}[section]

\newcommand{\bprob}[1]{\mathbb{P}\Bl[ #1 \Br]}
\newcommand{\prob}[1]{\mathbb{P}[ #1 ]}
\newcommand{\expect}[1]{\mathbb{E}[ #1 ]}
\newcommand{\bexpect}[1]{\mathbb{E}\Bl[ #1 \Br]}
\newcommand{\bbexpect}[1]{\mathbb{E}\lf[ #1 \ri]}
\newcommand{\vari}[1]{\text{VAR}[ #1 ]}
\newcommand{\bvari}[1]{\text{VAR}\Bl[ #1 \Br]}


\newcommand{\exming}{\ensuremath{\text{ExMin}_g} }
\newcommand{\imp}{\Longrightarrow}
\newcommand{\beq}{\begin{eqnarray*}}
\newcommand{\eeq}{\end{eqnarray*}}
\newcommand{\beqn}{\begin{eqnarray}}
\newcommand{\eeqn}{\end{eqnarray}}
\newcommand{\bemn}{\begin{multiline}}
\newcommand{\eemn}{\end{multiline}}

\newcommand{\sqeq}{\addtolength{\thinmuskip}{-4mu}
\addtolength{\medmuskip}{-4mu}\addtolength{\thickmuskip}{-4mu}}

\newcommand{\unsqeq}{\addtolength{\thinmuskip}{+4mu}
\addtolength{\medmuskip}{+4mu}\addtolength{\thickmuskip}{+4mu}}

% floor function
\newcommand{\lfl}{{\lfloor}}
\newcommand{\rfl}{{\rfloor}}
\newcommand{\floor}[1]{{\lfloor #1 \rfloor}}

\newcommand{\grad}[1]{\nabla #1}

% \newcommand{\sqeq}{\addtolength{\thinmuskip}{-4mu}
%   \addtolength{\medmuskip}{-4mu}\addtolength{\thickmuskip}{-4mu}}
% \newcommand{\unsqeq}{\addtolength{\thinmuskip}{+4mu}
%   \addtolength{\medmuskip}{+4mu}\addtolength{\thickmuskip}{+4mu}}

% \newcommand{\bea}{\begin{eqnarray}}
%   \newcommand{\eea}{\end{eqnarray}}
% \newcommand{\beas}{\begin{eqnarray*}}
%   \newcommand{\eeas}{\end{eqnarray*}}
% \newcommand{\bd}{\begin{displaymath}}
%   \newcommand{\ed}{\end{displaymath}}
% \newcommand{\be}{\begin{equation}}
%   \newcommand{\ee}{\end{equation}}
% \newcommand{\vs}{\vspace{0.2in}}
% \newcommand{\hs}{\hspace{0.5in}}
% \newcommand{\el}{\end{flushleft}}
% \newcommand{\bl}{\begin{flushleft}}
% \newcommand{\bc}{\begin{center}}
% \newcommand{\ec}{\end{center}}
% \newcommand{\remove}[1]{}

% \newtheorem{theorem}{Theorem}
% \newtheorem{corollary}{Corollary}
% \newtheorem{prop}{Proposition}
% \newtheorem{lemma}{Lemma}
% \newtheorem{defi}{Definition}
% \newtheorem{assum}{Assumption}
% \newtheorem{example}{Example}
% \newtheorem{property}{Property}
% \newtheorem{remark}{Remark}

% \newcommand{\separator}{
%   \begin{center}
%     \rule{\columnwidth}{0.3mm}
%   \end{center}
% }

% \newenvironment{separation}
% { \vspace{-0.3cm}
%   \separator
%   \vspace{-0.25cm}
% }
% {
%   \vspace{-0.5cm}
%   \separator
%   \vspace{-0.15cm}
% }

% \renewcommand{\baselinestretch}{0.98}

\def\A{\mathcal A}
\def\oA{\overline{\mathcal A}}
\def\S{\mathcal S}
\def\D{\mathcal D}
\def\eff{{\rm Eff}}

\def\cU{{\cal U}}
\def\cM{{\cal M}}
\def\cV{{\cal V}}
\def\cA{{\cal A}}
\def\cX{{\cal X}}
\def\cN{{\cal N}}
\def\cJ{{\cal J}}
\def\cK{{\cal K}}
\def\cL{{\cal L}}
\def\cI{{\cal I}}
\def\cY{{\cal Y}}
\def\cZ{{\cal Z}}
\def\cC{{\cal C}}
\def\cR{{\cal R}}
\def\id{{\rm Id}}
\def\st{{\rm st}}
\def\cF{{\cal F}}
\def\cG{{\cal G}}
\def\N{\mathbb{N}}
\def\R{\mathbb{R}}
\def\cB{{\cal B}}
\def\cP{{\cal P}}
\def\cS{{\cal S}}
%\def\and{\quad\mbox{and}\quad}
\def\ind{{\bf 1}}

\def\bmg{{\bm{\gamma}}}
\def\bmr{{\bm{\rho}}}
\def\bmq{{\bm{q}}}
\def\bmt{{\bm{\tau}}}
\def\bmn{{\bm{n}}}
\def\bmcapn{{\bm{N}}}
\def\bmrho{{\bm{\rho}}}

\def\igam{\underline{\gamma}(\lambda)}
\def\sgam{\overline{\gamma}(\lambda)}


\def\PP{{\mathrm P}}
\def\EE{{\mathrm E}}
\def\iskip{{\vskip -0.4cm}}
\def\siskip{{\vskip -0.2cm}}

\def\bp{\noindent{\it Proof.}\ }
\def\ep{\hfill $\Box$}


% \newcommand{\set}[1]{\mathcal{#1}}



